Lecturer
Frank de Zeeuw – email – Office: MA C1 557
If you have any questions, feel free to email me, or come by my office.
Course Details
- Lectures: Wednesday – 11:00-13:00 – MA A1 12
- Course book page
- This PhD course is worth 2 credits. To pass, you should attend most of the lectures, and give a presentation at the end. There will be some problem sets, but they will be optional. Master students are very welcome. There are no serious prerequisites for the course, aside from mathematical maturity and basic skills from combinatorics, algebra, and analysis.
Content
Arithmetic Ramsey theory:
- Arithmetic ramsey theorems of Hilbert, Schur, Van der Waerden, Rado, Hales-Jewett
- Roth’s Theorem on 3AP-free sets of integers
- The Density Hales-Jewett Theorem
- Three-term arithmetic progressions over finite fields and the slice rank method
Lecture notes for the first part of the course.
Sum-product bounds:
- Introduction to the sum-product phenomenon
- Solymosi’s sum-product bound over the reals and related results
- Elekes’s approach to sum-product bounds using point-line incidence bounds
- Rudnev’s point-plane incidence bound over finite fields and sum-product applications
Lecture notes for the second part of the course.
Schedule
Feb 22: Outline, Schur’s Theorem, Van der Waerden’s Theorem
March 1: Hales-Jewett Theorem and corollaries, bounds for Van der Waerden’s Theorem
March 8: Hilbert cubes, combinatorial proof of Roth’s Theorem
March 15: History, Behrend’s construction, regularity lemma proof of the Corners Theorem
March 22: The Density Hales-Jewett Theorem, part 1
March 29: The Density Hales-Jewett Theorem, part 2
April 5: The Density Hales-Jewett Theorem, part 3
April 12: Three-term arithmetic progressions over finite fields and the slice rank method
April 19: Break
April 26: The Erdős-Szemerédi sum-product theorem
May 3: Improved sum-product bounds of Elekes and Solymosi
May 10: Sum-product bounds over R using incidence geometry
May 17: Sum-product bounds over finite fields
May 24: Sum-product bounds over finite fields
May 31: Student presentations
Resources
There will be lecture notes to go with the lectures.
Here are some texts and sites that are relevant, although most of them only partially overlap this course.
- Leader – Ramsey theory (lecture notes)
- Tao – Arithmetic Ramsey theory (lecture notes)
- Schacht – Ramsey Theory (lecture notes)
- Graham-Rothschild-Spencer – Ramsey Theory (book)
- Gowers – Erdős and arithmetic progressions – (survey)
- Solymosi – Elementary additive combinatorics (lecture notes)
- Tao-Vu – Additive Combinatorics (book)
- Sheffer – Additive Combinatorics (course website)
- Trevisan – Additive combinatorics and theoretical computer science (survey)
Just for fun, here are some of the original papers of the theorems in this course:
Van der Waerden’s Theorem and related theorems:
- Hilbert – Ueber die Irreducibilität ganzer rationaler Functionen mit ganzzahligen Coefficienten (1892)
- Schur – Über die Kongruenz x^m+y^m=z^m mod p (1917)
- Van der Waerden – Beweis einer Baudetschen Vermutung (1927)
- Rado – Studien zur Kombinatorik (1933)
- Hales and Jewett – Regularity and positional games (1963)
Roth’s Theorem and related theorems:
- Behrend – On sets of integers which contain no three terms in arithmetical progression (1946)
- Roth – On certain sets of integers (1953)
- Szemerédi – On sets of integers containing no four elements in arithmetic progression (1969)
- Szemerédi – On sets of integers containing no k elements in arithmetic progression (1975)
- Solymosi – Note on a generalization of Roth’s theorem (2003)
The Density Hales-Jewett Theorem and related theorems:
- Ajtai and Szemerédi – Sets of lattice points that form no squares (1974)
- Furstenberg and Katznelson – A density version of the Hales-Jewett theorem (1991)
- Polymath – A new proof of the density Hales-Jewett theorem (2012)
- Dodos, Kanellopoulos, and Tyros – A simple proof of the density Hales-Jewett theorem (2014)
Arithmetic progressions over finite fields and the slice rank method:
- Ellenberg and Gijswijt – On large subsets of F_q^n with no three-term arithmetic progression (2017)
- Tao – A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound (2016)
- Naslund and Sawin – Upper bounds for sunflower-free sets (2016)
Sum-product bounds:
- Erdős and Szemerédi – On sums and products of integers (1983)
- Elekes – On the number of sums and products (1997)
- Elekes, Nathanson, and Ruzsa – Convexity and sumsets (1999)
- Solymosi – On distinct consecutive differences (2005)
- Solymosi – Bounding multiplicative energy by the sumset (2009)
Sum-product bounds over finite fields:
- Vinh – The Szemerédi–Trotter type theorem and the sum-product estimate in finite fields (2011)
- Rudnev – On the number of incidences between points and planes in three dimensions (2014)
- Roche-Newton, Rudnev, and Shkredov – New sum-product type estimates over finite fields (2016)
- Stevens and De Zeeuw – An improved point-line incidence bound over arbitrary fields (2016)
